On the first Steklov-Dirichlet eigenvalue on eccentric annuli in general dimensions

TL;DR

This paper extends the understanding of the first Steklov-Dirichlet eigenvalue on eccentric annuli from two dimensions to higher dimensions, using bispherical coordinates and Fourier-Gegenbauer series.

Contribution

It generalizes previous two-dimensional results to all dimensions by developing new analytical techniques involving bispherical coordinates and Fourier-Gegenbauer series.

Findings

Established the limiting behavior of the first eigenvalue in higher dimensions.

Extended previous two-dimensional results to general dimensions.

Developed a new analytical framework for eigenvalue problems in eccentric annuli.

Abstract

We consider the Steklov-Dirichlet eigenvalue problem on eccentric annuli in Euclidean space of general dimensions. In recent work by the same authors of this paper [21], a limiting behavior of the first eigenvalue, as the distance between the two boundary circles of an annulus approaches zero, was obtained in two dimensions. We extend this limiting behavior to general dimensions by employing bispherical coordinates and expressing the first eigenfunction as a Fourier-Gegenbauer series.